A Course in Minimal Surfaces

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Tobias Holck Colding; William P. Minicozzi II

Minimal surfaces date back to Euler and Lagrange and the beginning of the
calculus of variations. Many of the techniques developed have played key
roles in geometry and partial differential equations. Examples include
monotonicity and tangent cone analysis originating in the regularity
theory for minimal surfaces, estimates for nonlinear equations based on
the maximum principle arising in Bernstein's classical work, and even
Lebesgue's definition of the integral that he developed in his thesis on
the Plateau problem for minimal surfaces.

This book starts with the classical theory of minimal surfaces and ends up
with current research topics. Of the various ways of approaching minimal
surfaces (from complex analysis, PDE, or geometric measure theory), the
authors have chosen to focus on the PDE aspects of the theory. The book
also contains some of the applications of minimal surfaces to other fields
including low dimensional topology, general relativity, and materials
science.

The only prerequisites needed for this book are a basic knowledge
of Riemannian geometry and some familiarity with the maximum
principle.

Readership

Graduate students and research mathematicians interested in
the theory of minimal surfaces.

Minimal surfaces date back to Euler and Lagrange and the beginning of the
calculus of variations. Many of the techniques developed have played key
roles in geometry and partial differential equations. Examples include
monotonicity and tangent cone analysis originating in the regularity
theory for minimal surfaces, estimates for nonlinear equations based on
the maximum principle arising in Bernstein's classical work, and even
Lebesgue's definition of the integral that he developed in his thesis on
the Plateau problem for minimal surfaces.

This book starts with the classical theory of minimal surfaces and ends up
with current research topics. Of the various ways of approaching minimal
surfaces (from complex analysis, PDE, or geometric measure theory), the
authors have chosen to focus on the PDE aspects of the theory. The book
also contains some of the applications of minimal surfaces to other fields
including low dimensional topology, general relativity, and materials
science.

The only prerequisites needed for this book are a basic knowledge
of Riemannian geometry and some familiarity with the maximum
principle.